Abstract
Using the framework of formal theory of partial differential equations, we consider a method of computation of the bi-Hilbert polynomial (i.e. Hilbert polynomial in two variables). Furthermore, present an approach to compute the number of arbitrary functions of positive differential order in the general solution. Then, under the “AC=BD” model for mathematics mechanization developed by Hong-qing ZHANG, we present a method to reduce an overdetermined system to a well-determined one. As applications, the Maxwell equations and weakly overdetermined equations are considered.
Similar content being viewed by others
References
Ampère A M. Considérations Générales sur les Intégrales des Équations aux Différentielles Partielles. J École Polytéchnique, 1815, 10: 549–611
Einstein A. Appendix II. Generalization of Gravitation Theory. The Meaning of Relativity, 4th ed. Princeton: Princeton University Press, 1953, 135–165
Hausdorf M, Seiler W M. On the numerical analysis of overdetermined linear partial differential systems. Lecture Notes in Computer Science, 2003, 2630: 152–167
Hermann R. The Geometry of Non-Linear Differential Equations, Bäcklund Transformations, and Solitons, Part A. Interdisciplinary Mathematics XII. Brookline: Math Sci Press, 1976
Johnson J. Differential dimension polynomials and a fundamental theorem on differential modules. Amer J Math, 1969, 91: 239–248
Johnson J. Kahler differentials and differential algebra in arbitray characteristic. Trans Amer Math Soc, 1974, 192: 201–208
Kolchin E R. The notion of dimension in the theory of algebraic differential equations. Bull Amer Math Soc, 1964, 70: 570–573
Kolchin E R. Differential Algebra and Algebraic Groups. New York, Academic Press, 1973
Kondratieva M V. A minimal dimension polynomial of the extension of a field that is given by a system of linear differential equations. Translated from Matematicheskie Zametki, 1989, 45: 80–86
Kondratieva M V, Levin A B, Mikhalev A V, et al. Differential and Difference Dimension Polynomials. Dordrecht: Kluwer Academic Publishers, 1999
Kondratieva M V, Levin A B, Mikhalev A V, et al. Computation of dimension polynomials. Intern J Algebra Comput, 1992, 2: 117–137
Levin A. Computation of Hilbert polynomials in two variables. J Symbol Comput, 1999, 28: 681–710
Levin A. Reduced Gröbner bases, free difference-differential modules and difference-differential dimension polynomials. J Symbol Comput, 2000, 30: 357–382
Levin A. Multivariable dimension polynomials and new invariants of differential field extensions. Intern J Math Soc, 2001, 27: 201–214
Levin A. Multivariable difference dimension polynomials. J Math Sci, 2005, 131: 6060–6082
Pommaret J F. Systems of Partial Differential Equations and Lie Pseudogroups. London: Gordon & Breach, 1978
Reid G J. Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution. Eur J Appl Math, 1991, 2: 293–318
Saunders D J. The Geometry of Jet Bundles. London Mathematical Society Lecture Notes Series 142. Cambridge: Cambridge University Press, 1989
Seiler W M. Analysis and Application of the Formal Theory of Partial Differential Equations. Thesis submitted for the degree of Doctor of Philosophy, Lancaster University, 1994
Seiler W M. On the arbitrariness of the general solution of an involutive partial differential equation. J Math Phys, 1994, 35: 486–498
Seiler W M. Arbitrariness of the generral solution and symmetries. Acta Appl Math, 1995, 41: 311–322
Sit W. Differential dimension polynomials of finitely generated extensions. Proc Amer Math Soc, 1978, 68: 251–257
von Weber E. Partielle Differentialgleichungen. Enzyklopädie der mathematischen Wissenschaften, 1900, 2: 294–399
Xie F D, Zhang H Q. The dimension of the linear differential ideal. J Lanzhou University (Natural Sciences), 2003, 39: 4–7
Zhang H Q. The AC = BD model for mathematics mechanization. J Sys Sci & Math Sci, 2008, 28: 1030–1039
Zhang H Q, Ding Q. Analytic solutions of a class of nonlinear partial differential equations. Appl Math Mech, 2008, 29: 1–12
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ding, Q., Zhang, H. Arbitrariness of the general solution of the partial differential equations and its applications. Sci. China Math. 53, 1731–1741 (2010). https://doi.org/10.1007/s11425-010-3011-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-3011-1